What is required of me for not being able to stone the Jamarat on the third day on behalf of my mother, after I had deputized for her in stoning the Jamarat on the first and second days?

You and your mother are required to offer a sacrifice (a dam to be slaughtered in the Haram (definition) and distributed among the poor) for the last day on which you did not stone the Jamarat (definition). Another dam is required for not spending the night in Mina if you departed before most of the night had passed. It would have been better for you to stone the Jamarat on foot, even if the car roads were closed.